I understand space-time to be like a coordinate system. Am I misunderstanding?
A coordinate system is like a mapping plan. Variations in the density of matter (and energy) change the landscape from something, say, flat to regions with dimples, large and small. The analogy of putting a bowling ball on a trampoline is often used where the depression in the rubber sheet represents how the map, or coordinate system, can be used to properly take us from a simple, and incorrect, Cartesian X-Y grid and give us something that can help us produce correct results, such as acceleration rates through it. The spacing between the lines drawn represent the gradient -- the change in the rate of lines encountered with distance.
The mapping of the region around a small black hole, for instance, would reveal a severe gradient the closer one got to the EH. So, if someone were falling in this gradient would be so severe it would pull the body apart - spaghettification.
A GR physicist, however, I'm not, but this is my interpretation of the more basic view of it.